Don’t confuse the fact an h is “aspirated” or not, and the fact an h is pronounced or not.

Even when not pronounced like in standard modern French, the aspirated h still exists because of its side effects.

For example homme has a non aspirated h so is pronounced exactly like if written omme thus un homme is almost pronounced like “un nomme“, because of the liaison.

There is such an h in the English hour (An hour); compare with house (A house).

In house, the h is voiced as a consonant. This consonant no more exists¹ in French. When the h is aspirated, it just prevents the liaison to be realized:

Un héros is pronounced like “un éro“, never “un néro“.

Why did the French stop pronouncing the h sound?

I guess because dropping it was a simplification that wasn’t breaking the comprehensibility.

Does the bolded part mean that those words which are today pronounced with h aspiré were then pronounced with h muet?

No, that means aspirated h ceased again to be voiced.

See also: How can we distinguish "H-muet" or "H-aspiré"? and Understanding an extract on h-aspiré

¹_{We still pronounce the h sound in a small set of words. See the linked questions.}